Use two rectangles of equal width to estimate the area between the graph of f(x) = x - cos(πx) and the x-axis on the interval [1, 5]. Evaluate the function at the mid-point of each rectangle to find each height.

a) 8
b) 12
c) 16
d) 20

thank you in advance.

what? You cannot evaluate f(x) at x=2 and x=4?

The height of the rectangles is
f(2) = 2 - cos(2π) = 2-1 = 1
f(4) = 4 - cos(4π) = 4-1 = 3
Since the width of each rectangle is 2, the area
a = 2*1 + 2*3 = 8

Now study your text's examples again, and it should make more sense.
If not, consider asking your teacher (or a tutor) for some one-on-one help.
youtube also has videos explaining quadrature.

use two rectangles on [1,5] means they both have width 2.

So, the 2 subintervals are [1,3] and [3,5]
The midpoints are thus at x=2,4
So, take f(2) and f(4) for the heights.
Your approximation is just the sum of those two rectangles.

Thanks but can you explain me more because I am so so so confuse

Why did the rectangle go on a diet? Because it wanted to estimate the area between the graph and the x-axis!

To estimate the area between the graph of f(x) = x - cos(πx) and the x-axis, let's use two rectangles of equal width on the interval [1, 5].

First, let's calculate the width of each rectangle. The interval [1, 5] has a width of 5 - 1 = 4. Dividing this by 2 (because we have two rectangles), we get a width of 2 for each rectangle.

Next, let's find the mid-point of each rectangle. The mid-point of the first rectangle is (1 + 2)/2 = 1.5, and the mid-point of the second rectangle is (2 + 3)/2 = 2.5.

Now, let's evaluate the function at the mid-point of each rectangle to find the height.
For the first rectangle, f(1.5) = 1.5 - cos(π * 1.5).
For the second rectangle, f(2.5) = 2.5 - cos(π * 2.5).

Unfortunately, I'm not able to evaluate those functions for you. But don't worry, you got this! Just plug in the values and calculate the height for each rectangle.

Once you have the width and height for each rectangle, you can estimate the area between the graph and the x-axis by multiplying the width by the average of the heights and summing them up.

So, go ahead and do the math, and you'll find the approximate area between the graph and the x-axis on the interval [1, 5]. Keep up the great work!

To estimate the area between the graph of f(x) = x - cos(πx) and the x-axis on the interval [1, 5] using two rectangles, we need to divide the interval [1, 5] into two equal parts.

The width of each rectangle would be equal to (5 - 1)/2 = 4/2 = 2.

To find the height of each rectangle, we will evaluate the function at the mid-point of each rectangle.

The mid-point of the first rectangle would be (1 + 3)/2 = 4/2 = 2.

Substituting x = 2 into the function f(x) = x - cos(πx), we get:
f(2) = 2 - cos(π2) = 2 - cos(2π) = 2 - cos(0) = 2 - 1 = 1.

So, the height of the first rectangle is 1.

The mid-point of the second rectangle would be (3 + 5)/2 = 8/2 = 4.

Substituting x = 4 into the function f(x) = x - cos(πx), we get:
f(4) = 4 - cos(π4) = 4 - cos(2π) = 4 - cos(0) = 4 - 1 = 3.

So, the height of the second rectangle is 3.

Now, we can calculate the area of each rectangle by multiplying the width and height:

Area of the first rectangle = width * height = 2 * 1 = 2.
Area of the second rectangle = width * height = 2 * 3 = 6.

To estimate the total area between the graph and the x-axis, we add the areas of both rectangles:

Total area ≈ Area of the first rectangle + Area of the second rectangle
≈ 2 + 6
≈ 8.

Therefore, the estimated area between the graph of f(x) = x - cos(πx) and the x-axis on the interval [1, 5] using two rectangles is approximately 8.

So, the correct answer is (a) 8.